Approxy of Born — Wikipedia

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Born approximation scheme in the first order: only the incident wave and the incident wave having interacted only once in all points R ‘of the diffusion potential are considered in the dissemination in r.

L’ approximation de Born is an approximation made in theory of diffusion, in particular in quantum mechanics, for potential very little dense diffusers. The approximation of Born to the first order consists in taking into account only the incident wave and waves disseminated by a single interaction with the potential in the description of the Total Total Wave [ first ] . She is named after Max Born.

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This is the disturbance method applied to diffusion on an extensive body.

Born approximation is used in many situations in physics.

In the dissemination of neutrons, the approximation of Born to the first order is almost always adequate, with the exception of phenomena of neutronic optics such as total internal reflection in a neutron guide, or of low -incidence dissemination and low angles.

We define the operator of the Green function, where

ϵ {displaystyle epsilon }

is an infinitesimal quantity:

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G^± ≡ G^O ( AND ± i ϵ ) = first E−Ho±iϵ{displaystyle {widehat {G}}_{pm }equiv {widehat {G}}_{o}(Epm iepsilon )={frac {1}{E-H_{o}pm iepsilon }}}

G ± ( r , r ′ ) ≡ ℏ22m⟨ r | G^± | r ′ ⟩ = – e±ik|r−r′|4π|r−r′|{Displaystyle G_{PM }(Mathbf {R} ,Mathbf {R} , ‘frac {frac {hbar ^{2m}}{2m}}{R}}}Langle Mathbf {R} |{Widehat {g}}_{PM }|mathbf {R} ‘Rangle =-{Frac {E^{PM IK|mathbf {R} -Mathbf {R} ‘|” Mathbf {R} -Mathbf {R} ‘||}}}

G ± ( r , r ′ ) ≈ |r|≫|r′|– e±ik|r|e∓ikr^⋅r′4π|r|= – e±ikre∓ik′⋅r′4πr{Displaystyle G_{PM }(Mathbf {R} ,Mathbf {R} , ‘Underset {|Mathbf {R} |GG |mathbf {R} Mathbf {R} |}E^{mp ikmathbf {hat {r}} cdot mathbf {R} ’}{4Pi |mathbf {r}=-{frac {e^{pm ikr}E^{mp Imathbf {K} ‘cdot mathbf {r} , {4Pi r}}}

The demonstrations of these relationships are found in the book Modern Quantum Mechanics by J. J. Sakurai (in) [ 2 ] as well as in the book Quantum mechanics II de Claude Cohen-Tannoudji [ 3 ] .

L’umquation the lippmann-swing:

| Φ ( ± ) ⟩ = | ϕ ⟩ + G^± IN | Φ ( ± ) ⟩ {Displayystyle | Psi ^{(PM)} Rangle = | Phi Rangle +{WideHat {g}} _ {PM} V | PSI ^{(PM)} Rangle}

Or

| ϕ ⟩ {Displaystyle | Phi Rangle}

is a solution of Schrödinger’s equation for a free particle. We will take the flat wave solution

| ϕ p⟩ = ℏ – 3 /2 | ϕ k⟩ {DISPLAYSTYLE | PHI _ {MOOBF {P} Rangle = Harb {- 3/2} | Phb-{k} Rangle} Rangle}

expressed respectively depending on the momentum

p {displaystyle p}

and the propagation vector

k {displaystyle mathbf {k} }

. When you express it all in the base of the position

⟨ r | {displaystyle langle mathbf {r} |}

, on a :

⟨r|ψ(±)⟩=⟨r|ϕp⟩+⟨r|G^±V|ψ(±)⟩=⟨r|ϕp⟩+∫d3r′⟨r|G^±|r′⟩⟨r′|V|ψ(±)⟩{Displaystyle {Begin{Aligned}{2}Langle Mathbf {R} |psi ^{(pm )}rangle &=langle Mathbf {R} |Phi {mathbf {p} Rangle +Langle Mathbf {R} |{Widehat {g}}_{pm }V|psi ^{(PM )}Rangle \&=Langle Mathbf {R} |PHI _{Mathbf {P} }Rangle +int D^{3Langle Mathbf {R} |{widehat {g}}_{pm }|mathbf {R} ‘Rangle Langle Mathbf {R} ‘|V|psi ^{(PM )} Rangle End{Aligneedat}}}}

In the case of a local V potential V, where

⟨ r ′ | IN | r ″ ⟩ = IN ( r ′ ) d 3 ( r ′ – r ″ ) {Displaystyle Langle Mathbf {R} ‘|V|mathbf {R} ‘Rangle =V(Mathbf {R} ‘)Delta ^{3}(Mathbf {R} ‘-Mathbf {R}”

:

⟨r|ψ(±)⟩=⟨r|ϕp⟩+∫d3r′⟨r|G^±|r′⟩V(r′)⟨r′|ψ(±)⟩≈⟨r|ϕp⟩−2mℏ214πe±ikrr∫d3r′e∓ik′⋅r′V(r′)⟨r′|ψ(±)⟩=(12π)3/2eik⋅r−(12π)3/2e±ikrr2mℏ2(2π)34π∫d3r′e∓ik′⋅r′(2π)3/2V(r′)⟨r′|ψ(±)⟩=(12π)3/2{eik⋅r−e±ikrr2mℏ2(2π)34π⟨ϕk′|V|ψ(±)⟩}{Displaystyle {Begin{Aligned}{2}Langle Mathbf {R} |psi ^{(pm )}rangle &=langle Mathbf {R} |Phi {mathbf {p} Rangle +int d^{3}r’ Langle Mathbf {R} |{Widehat {g}}_{PM }|mathbf {R} ‘Rangle V(Mathbf {R} ‘)Langle Mathbf {R} ‘|psi ^{(pm )}Rangle \&approx Langle Mathbf {R} |Phi _{Mathbf {P} }Rangle -{frac {2m}{HBAR ^{2}}{frac {1}{4Pi }{frac {e^{pm ikr}}{r}{r}{r}{r}{r} Int D^{3}R’E^{mp Imathbf {K} ‘Cdot Mathbf {R} ‘}V(Mathbf {R} ‘)Langle Mathbf {R} ‘|psi ^{(PM )}Rangle \&= Left( {Frac {1}{2Pi }}E^{Imathbf {K} CDOt Mathbf {R}-Left }-Left({Frac {1}{2Pi }} Right)^{3/ 2}{frac {e^{pm ikr}}{r}}{frac {2m} {hbar ^{2}}{frac {(2Pi )}{3Pi }}{4Pi }}} int d^{3} r'{frac {e^{mp imatbf {k} ‘cdot mathbf {r} ’}{left(2Pi right)^{3/2}}}}}}V(mathbf {r} ‘)langle mathbf {R} ‘ |psi ^{(pm )}Rangle \&=Left({Frac {1}{2Pi }} Right)^{3/2}{E^{E^{E^{E^{E^{E^{EKA {K} CDOt Mathbf {R}-{EFRAC {E ^{PM IKR}}{R}}{FRAC {2M}}{HBAR ^{2}}}{FRAC {(2PI )^{3}}{4PI }}Langle Ph _{Mathbf {K} ‘}|V |psi ^{(pm )}Rangle }ent{alignedat}}}

To better interpret the different terms, we can rewrite as follows:

⟨ r | Φ ( ± ) ⟩ = ( 12π) 3 /2 { It is i k⋅ r– e±ikrr f ( k ′ , k ) } {Displaystyle Langle Mathbf {R} |psi ^{(PM )}Rangle =Left({Frac {1}{2Pi }}^{3/2}{3/2} {KI IMATHBF {K} CDOt Mathbf {R} } -{frac {e^{pm ikr}}{r}}F(mathbf {k} ‘,mathbf {K}}}}

Or

f ( k ′ , k ) {displaystyle f(mathbf {k} ‘,mathbf {k} )}

is called “the amplitude of diffusion”. The first term still represents the incident wave in the direction

k ≡ piℏ {displaystyle mathbf {k} equiv {frac {mathbf {p_{i}} }{hbar }}}

While the form of the second term is interpreted as an outgoing spherical wave in the case

⟨ r | Φ ( + ) ⟩ {DisplayStyle Langle Mathbf {R} | Psi ^{(+)} Rank}

and entering the case

⟨ r | Φ ( – ) ⟩ {DisplayStyle Langle Mathbf {R} | Psi ^{(-)} Ranle}

. To this point, however,

f ( k ′ , k ) {displaystyle f(mathbf {k} ‘,mathbf {k} )}

is expressed in terms of

| Φ ( ± ) ⟩ {displaystyle |psi ^{(pm )}rangle }

, potentially unknown. We therefore seek to re-express it in known terms, such as

| ϕ k⟩ {displaystyle |phi _{mathbf {k} }rangle }

and V, and that is the whole point of Born’s approximation.

We multiply the equation of Lippman-Schwinger by the Diffuser V:

IN | Φ ( ± ) ⟩ = IN | ϕ k⟩ + IN G^± IN | Φ ( ± ) ⟩ {Displaystyle v | psi ^ ^ ^{(pm)} rangle = v | pi _ {mathbf {k}} rangle +v {Widehat {G}}} v | psi ^{(pm) ^{(pm)} rangle}

We replace it in the Lippman-Schwinger equation, we reiterate if necessary, ultimately approximating to the desired order in V:

V|ψ(±)⟩=V|ϕk⟩+VG^±V(|ϕk⟩+G^±V|ψ(±)⟩)=V|ϕk⟩⏟1erordre+VG^±V|ϕk⟩⏟2eordre+VG^±VG^±V|ψ(±)⟩=V|ϕk⟩+VG^±V|ϕk⟩+VG^±VG^±V|ϕk⟩⏟3eordre+VG^±VG^±VG^±V|ψ(±)⟩{Displaystyle {BEGIN {alignedat} {2} v | PSI ^{(pm)} rangle & = v | phi _ {mathbf {k}} rangle +v {widehat {g} _ {} v (| Phi _ {mathbf {k}} rangle +{widehat {g}} _ {pm} v | psi ^{(pm)} rangle) \ & underbrace {underbrace {v | phi _ {mathbf {k}} rangle} _ { 1^{er} ordre}+v {widehat {g}} _ {pm} v | phi _ {mathbf {k}} rangle} _ {2^{e} ordre}+v {widehat {g} _ {{ PM} v {widehat {g}} _ {pm} v | psi ^{(pm)} rangle \ & = Underbrace {v | phi _ {mathbf {k}}}} {pm } V | phi _ {mathbf {k}}} rangle +v {widehat {g}} _ {pm} v {widehat {g} _ {pm} v | phi _ {mathbf {k}}} _ {3 ^{e} ordre}+v {widehat {g}} _ {pm} v {widehat {g}} _ {pm} v {widehat {g} _ {pm} v | psi ^{(pm)} rangle End {alignedat}}

By replacing in the expression of

f ( k ′ , k ) {displaystyle f(mathbf {k} ‘,mathbf {k} )}

, we therefore have a decomposition of this one:

f(k′,k)=∑i=1∞f(i)(k′,k)f(i)(k′,k)=−2mℏ2(2π)34π⟨ϕk′|V(G^±V)(i−1)|ϕk⟩{Displaystyle {Begin{Alignedat}{2}FF(MATHBF {K} ‘,Mathbf {K} )&=Sum _{I=1}F{(I)}(Mathbf {K} ‘, Mathbf {K} )\F^{(I)}(Mathbf {K} ‘,Mathbf {K} )&=-{FRAC {2M}}{HBAR ^{2}}{frac {(2PI )^{3 }}{4PI }}Langle PHI _{Mathbf {K} , , ‘}|v({Widehat {g}}_{pm }^{(i-1)}|Phi _{mathbf {k} }rangle end {Alignedat}}}

  1. ↑ C. Cohen-tanudji, B. Diu et F. Laloë, Quantum mechanics [Edition detail] , vol. 2, [Paris] Hermann, 1993, ©1973 (ISBN  978-2-70566121-2 ) , p. 911.
  2. ↑ Sakurai 1994, p. 381.
  3. ↑ COHEN-ANNOUDJI, says Et Laloë 1997, p. 906-908.

Document utilisé pour la rédaction de l’article: document used as a source for writing this article.

  • (in) Jun John Sakurai , Modern Quantum Mechanics , Reading (Mass.), Addison Wesley, , 500 p. (ISBN  978-0-201-53929-5 , BNF  39112504 , Online presentation ) . Document utilisé pour la rédaction de l’article
  • (in) Ta-you Wu et takashi Ohmura , Quantum Theory of Scattering , Prentice Hall, ( Online presentation ) . Document utilisé pour la rédaction de l’article
  • (in) John Robert Taylor , Scattering Theory : The Quantum Theory of Nonrelativistic Collisions , Wiley, , 477 p. (ISBN  978-0-471-84900-1 And 9780471849001 )
  • Claude Cohen-Tannoudji , Bernard Says and Franck Laloë , Quantum mechanics II , Hermann,

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